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Full text of "Mathematical And Physical Papers - Iii"

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In order that the right-hand members of th£se equations may be perfect differentials, we must have
dydz^  '        diidx~~  '        dxdy d*8    d2S               d*S    d*S _ 0         d?S    cPS^.__ ~
and therefore
The equations (/), (#) give
7CV            7 fs                        /JSJ
so that ~- ,  -T- i  and -7- are constant.    Substituting in (e) and cfo    ay          d#
integrating, and then substituting in (138) the resulting expressions for G)', a)", a/", and inuegrating again, we shall obtain the results given in Art. 50.
[The possibility of a more general kind of motion than that of a solid taking place in an elastic fluid without consumption of energy by internal friction, that is, without its. being converted into the kinetic energy of heat, depends on the coefficient in the last term of (1) being /n/3, or rather not greater than /x/3 ; and that again on the assumption made in a former paper (Vol. I. p. 87) that in any elementary portion of the fluid a velocity of dilatation alike in all directions does not affect the hydrostatical relation between the pressure and density. Although I have shown (Vol. I., p. 119) that on the admission of a supposition which Poisson would probably have allowed the two constants in his equations of motion are reduced to one, and the equations take the form (1), and although Maxwell obtained the same equations from his kinetic theory of gases (Philosojriiical Transactions for 1867, p. 81) I have always felt that the correctness of the value //,/3 for the coefficient of the last term in (1) does not rest on as firm a basis as the correctness of the equations of motion of an incompressible fluid, for which the last term does not come in at all. If the supposition made above be not admitted, we must replace the coefficient /Lt/3 by a different coefficient, which